The Spectrum of Hyperbolic Surfaces

By: Nicolas Bergeron

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Synopsis: Thistext is an introduction to the spectral theory of the Laplacian oncompact or finite area hyperbolic surfaces. For some of thesesurfaces, called "arithmetic hyperbolic surfaces", theeigenfunctions are of arithmetic nature, and one may use analytictools as well as powerful methods in number theory to study them. Afteran introduction to the hyperbolic geometry of surfaces, with aspecial emphasis on those of arithmetic type, and then anintroduction to spectral analytic methods on the Laplace operator onthese surfaces, the author develops the analogy between geometry(closed geodesics) and arithmetic (prime numbers) in proving theSelberg trace formula. Along with important number theoreticapplications, the author exhibits applications of these tools to thespectral statistics of the Laplacian and the quantum uniqueergodicity property. The latter refers to the arithmetic quantumunique ergodicity theorem, recently proved by Elon Lindenstrauss. Thefruit of several graduate level courses at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces allows the reader to review an array of classical results andthen to be led towards very active areas in modern mathematics.